You make it look as if one should stay exactly on the Lagrangean point to appreciate its benefits.

The reality is that spacecrafts fly on very large orbits around these points: the Planck spacecraft followed a 400,000 km-wide orbit [1] around the Sun-Earth L₂ point, and this is the same for many other spacecrafts that have flown around that place.

400,000 km is ~30 times larger than the Earth's diameter. Unless somebody has nasty intentions like e.g. purposefully crashing their own spacecraft against some other, there is no reason to be worried. There is more space around these points than here on Earth.

[1] https://sci.esa.int/web/planck/-/34728-orbit-navigation

You don't need to reach the exact Lagrange point and reduce your velocity relative to it to 0. James Webb will be orbiting the Sun-Earth L2 with an apogee of nearly a million miles. There is a ridiculous amount of room for craft to orbit L2, just like there is to orbit Earth (beyond LEO).

Also there are other satellites at L2 already.

https://en.wikipedia.org/wiki/List_of_objects_at_Lagrange_po...

It's really a none issue

The United States Air Force started off as a division of the United States Army Signal Corps. It didn't become its own branch of the military until after 40 years and two world wars.

Lack of a "space force" doesn't mean lack of military capability in space.

I don't think there's much ambiguity around a military's utility in forcefully enclosing the commons, or claiming extranational resources at gunpoint. I suspect people who are leery of a 'Space Force' have other reasons to feel that way. And I'd hope there are other ways of managing the sharing of things like Lagrange points than outright violence (though I suppose some body must have the capability to employ force or else any agreements made would be easily violated, yadda yadda).

How would a space force take possession of a lagrage point, though, without creating a permanent field of dangerous debris?

AFAIK Lagrange points are not fully stable, so any debris cloud will likely disperse into the Solar orbit, which is kinda big.

None of the ideas in the sibling comments (to my post) will work, because they ignore the fact that enemies will employ counter-measures which may cause debris to be formed. In the extreme case, a nation's military may decide to deny the area to anyone rather than lose their own access, intentionally creating such a debris field.

You'd have to fill the volume of at 25,000 Earths (or more, that's just my napkin math), where orbits aren't stable for more than a year or two, with enough debris to hit something a few meters across. No using explosions to do this either because debris with more than a few m/s aberrant velocity would be ejected from the Lagrange point even faster. Seems unlikely.

The entire thing is unlikely. I'm just pointing out that you can't claim space 'cleanly' when your adversaries are actively opposing your claim. To play devil's advocate further though:

As the 'defender' you likely don't need stable orbits. A year or two of denial might buy you what you want. You also don't need to ensure a collision against a single small target, you just need enough debris to make use of the area cost prohibitive in terms of risk.

Rendezvous, attach booster, and deorbit, or just fly away?

Only reasonable way at the moment would be some kind of satellite capture system that nobody has publicly demonstrated.